MISC

Euler systems for Galois deformations and the pseudo-isomorphism class
of the dual of fine Selmer groupsTatsuya Ohshita, 2017年12月26日, In this article, we study the pseudo-isomorphism class of the dual fine
Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation
ring $\Lambda$ is isomorphic to the ring of formal power series. By using the
"Kolyvagin system" arising from a given Euler system $c$, we shall construct
ideals $C_i(c)$ of $\Lambda$, and prove that the ideals $C_i(c)$ approximate
the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we
shall prove that the ideals $C_i(c)$ arising from the Euler system of
Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual
fine Selmer groups attached to ordinary and nearly ordinary Hida deformations
satisfying certain conditions.

On higher Fitting ideals of Iwasawa modules of ideal class groups over
imaginary quadratic fields and Euler systems of elliptic units IITatsuya Ohshita, 2017年10月13日, In our previous work, by using Kolyvagin derivatives of elliptic units, we
constructed ideals C_i of the Iwasawa algebra, and proved that the ideals C_i
become "upper bounds" of the higher Fitting ideals of the one and two variable
p-adic unramified Iwasawa module X over an abelian extension field K_0 of an
imaginary quadratic field K. In this article, by using "non-arithmetic"
specialization arguments, we prove that the ideals C_i also become "lower
bounds" of the higher Fitting ideals of X. In particular, we show that the
ideals C_i determine the pseudo-isomorphism class of X. Note that in this
article, we also treat the cases when the p-part of the equivariant Tamagawa
number conjecture (ETNC)_p is not proved yet.
In the cases when (ETNC)_p is proved, stronger results have already been
obtained by Burns, Kurihara and Sano: under the assumption of (ETNC)_p and
certain conditions on the character $\psi$ on the Galois group of K_0/K, they
have given a complete description of the higher Fitting ideals of the
\psi-component of X by using Rubin--Stark elements. In our article, we also
prove that the $\psi$-part of C_i coincide with the ideals constructed by
Burns, Kurihara and Sano in certain cases when (ETNC)_p is proved. As a
corollary of this comparison results, we also deduce that the annihilator ideal
of the $\psi$-part of the maximal pseudo-null submodule of X coincides with the
initial Fitting ideal in certain situations.